b1 =1r11 r2 - university of california, san...

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR

AN EQUATION FOR BEHAVIORAL CONTRAST

BEN A. WILLIAMS AND JOHN T. WIXTED

UNIVERSITY OF CALIFORNIA, SAN DIEGO

Pigeons were trained on a three-component multiple schedule in which the rates of reinforcement inthe various components were systematically varied. Response rates were described by an equationthat posits that the response-strengthening effects of reinforcement are inversely related to the contextof reinforcement in which it occurs, and that the context is calculated as the weighted average of thevarious sources of reinforcement in the situation. The quality of fits was comparable to that foundwith previous quantitative analyses of concurrent schedules, especially for relative response rates,with over 90% of the variance accounted for in every case. As with previous research, reinforcementsin the component that was to follow received greater weights in determining the context than didreinforcements in the preceding component.Key words: multiple schedules, quantitative analysis, behavioral contrast, inhibition of reinforce-

ment, matching, pigeons

Response rate in a multiple schedule is afunction of relative rate of reinforcement, atleast to a first approximation (Williams,1983a). Several theoretical analyses havespecified this relationship in quantitativeterms, in the hope of more clearly delineatingthe underlying principles of schedule inter-actions. One approach is the generalizedmatching law (Baum, 1974; Lander & Irwin,1968), as given by Equation 1, in which B1represents response rate in component 1, B2represents response rate in component 2, R,and R2 the corresponding reinforcement rates,b the bias toward one component or the other,and a the sensitivity of the relative responserates to the distribution of reinforcement:

B1 =1R11B2 R2 (1)

Although Equation 1 has been shown todescribe interactions in multiple schedules withconsiderable accuracy (see Table 1 of Wil-liams, 1983a), it has the limitation of describ-

This research was supported by NIMH research grant1 R01 MH 35572-02 and NSF grant BNS 84-08878 tothe University of California. Portions of this research werereported at the 1982 meeting of the Psychonomics Society.The authors thank M. Davison, E. Fantino, R. J. Herrn-stein, and J. A. Nevin for their helpful comments onprevious versions of the manuscript.John T. Wixted is now at Emory University. Requests

for reprints should be addressed to Ben A. Williams, De-partment of Psychology, C-009, University of Californiaat San Diego, La Jolla, California 92093.

ing only relative response rates, with no ob-vious method of extension to absolute ratesthat presumably are more fundamental.Moreover, McLean and White (1983) haveshown that the degree of sensitivity of relativeresponse rates to the reinforcement distribu-tion may be totally independent of contrastinteractions between the two components ofthe multiple schedule. That is, a high valueof a may result from a large amount of alter-native reinforcement that competes with theoperant response within a given component,and thus may occur in the complete absenceof behavioral contrast. To the extent that anaccount of contrast is a major concern, there-fore, some alternative formulation seems de-sirable.

Herrnstein (1970) proposed Equation 2,derived from his description of response ratesin concurrent and simple variable-interval (VI)schedules. The terms of Equation 2 corre-spond to those of Equation 1 with the additionof Ro corresponding to reinforcement in thesituation other than that scheduled by the ex-perimenter, and with k corresponding to theasymptotic response rate. Equation 2 capturesthe degree of interaction between the compo-nents by the parameter m, which is assumedto vary between 0 and 1 as a function of thevariables that determine the degree of inter-action (typically the duration of the schedulecomponents). The derivation of relative re-sponse rates is then given by writing Equation2 for each separate component as shown inEquation 3:

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1986, 45, 47-62 NUMBER 1 (JANUARY)

BEN A. WILLIAMS and JOHN T. WIXTED

B1= k R (2)

ki RIB1 1 R, + mR2 +RJ?3R= mR+R (3)B2 k- R2

R2 + mR1 + RoAlthough Equations 2 and 3 do have the

advantage of specifying the relation betweenrelative and absolute response rates, they alsomake several incorrect predictions, as noted byseveral investigators (de Villiers, 1977;McLean & White, 1983; McSweeney, 1980;Spealman & Gollub, 1974). Perhaps the mostserious of these is the prediction that absoluteresponse rates during both components willdecrease when component duration is short-ened, whereas in fact the response rate in thecomponent with the higher reinforcement ratetypically increases (Williams, 1980). A relat-ed problem is that Equation 2 predicts no con-trast after a transition from a simple VI to amultiple VI extinction schedule, and, con-versely, predicts that there should be a de-crease in response rate when a simple VI ischanged to a multiple VI VI with equal ratesof reinforcement. The reason for both predic-tions is that any increase in the denominatorof Equation 2 (either by increasing m or byincreasing R2) should decrease the responserate.McLean and White (1983) also have ar-

gued that Equation 2 is conceptually incon-sistent with the matching law upon which itwas originally based. According to Herrnstein(1970, 1974), the parameter k in Equation 2refers to the total amount of behavior possiblein the situation, and should depend only onthe units of measurement. Thus, during Com-ponent 1 of a multiple schedule, the only be-havior possible is the key pecking under thecontrol of the particular discriminative stim-ulus plus the other activities that are main-tained by the reinforcers not scheduled by theexperimenter (RO). Similarly, during Com-ponent 2, the behavior possible is pecking oc-casioned by the second discriminative stimulusand, again, the alternative activities main-tained by R.. Letting Bo correspond to Ro,therefore, the sum of B1 and Bo and the sumof B2 and Bo should both equal k. However,this cannot be true if the value of the rein-forcement in the alternative component has an

effect. Thus, to the extent that successive in-teractions occur, they cannot be explainedwithin the matching framework proposed byHerrnstein (but see Staddon, 1982, andMcLean & White, 1983, for alternative for-mulations based on the matching law).

Equations 1 and 2 both assume that relativerate of reinforcement is a molar variable,which cannot be reduced to more elementaryprocesses. This assumption is challenged byseveral demonstrations that contrast interac-tions are temporally asymmetric when mul-tiple schedules with more than two compo-nents are investigated (Farley, 1980; Williams,1976a, 1976b, 1979, 1981). That is, varia-tions in relative rate of reinforcement haveproduced only weak interactions when varia-tion was due to changes in the precedingschedule, but have produced much larger in-teractions when the variation was due tochanges in the following schedule. Because thetwo effects appear to be functionally separate,and because both presumably are operative inthe typical two-component schedule, the im-plication is that relative rate of reinforcementin the typical schedule cannot be a unitaryvariable.

Despite the problem of two effects operat-ing simultaneously, results from other exper-imental areas offer strong encouragement forretaining relative rate of reinforcement as afundamental concept. For example, Fantino's(1977) delay-reduction account of choice inconcurrent-chains schedules assumes that theconditioned reinforcing properties of a stim-ulus are a function of its average time to pri-mary reinforcement, relative to the averagetime to primary reinforcement in the situationas a whole. Similarly, Gibbon's (1981) ac-count of autoshaping assumes that the rate ofconditioning produced by a given signal-rein-forcer relation is always relative to the inter-reinforcement interval in the situation inde-pendent of the signal. The considerable successof such formulations, and their independentdevelopment, suggests that "context of rein-forcement" is a fundamental concept that mustbe considered for any conditioning situation.An alternative conceptualization of context

of reinforcement with respect to multipleschedules comes from an extension of Catan-ia's (1973) account of behavior in single-schedule situations, based on his concept of"inhibition of reinforcment" (see also Keller,

48

ANALYSIS OF CONTRAST

1980). Catania argued that reinforcement hastwo effects: increasing the strength of the re-sponse on which it is contingent according toEquation 4, and "inhibiting" all behavior inthe situation according to Equation 5. Notethat K is in units of responses/reinforcers andC is in units of reinforcers/time, and that MRrefers to all sources of reinforcement in thesituation (and is equal to R1 when only a sin-gle reinforcement schedule is presented):

f(E) = KR1; (4)f(I) 1 (5)

zR1+ C

Combining the two expressions produces thefull equation for response rate,

B KCR,1 ZR + C'

which becomes Equation 6 when the com-bined parameters KC are set equal to a newconstant, s.

B1=ZR+C (6)

Note that this implies that s is in units ofresponses/time and that its estimation will bedirectly correlated with estimates of C. Alsonote that Equation 6 is similar to Herrnstein's(1970) account of behavior in simple sched-ules, but the interpretation of parameters isdifferent.

Catania's (1973) concept of inhibition byreinforcement can be extended to multipleschedules by assuming that the amount of suchinhibition is a function of the average rate ofreinforcement in the situation. However, asimple arithmetic average is inappropriate,given that different sources of reinforcementproduce different degrees of interaction. Thus,it is necessary to use a weighted average inwhich different weights are assigned to thedifferent sources of reinforcement. Equation7 captures this idea, with Bn referring to theresponse rate in component n, Rn to the re-inforcement rate in that component, Rn-1 tothe reinforcement in the preceding compo-nent, Rn+1 to the reinforcement in the follow-ing component, p and f to the weights for thepreceding and following sources of reinforce-

ment relative to that in the prevailing com-ponent (which is always 1.0), and s and C tothe same as in Equation 4:

B =s-nn Rn + pRn_1 + fRn+IC

1 +p +f(7)

The rationale underlying Equation 7 is thatcontrast occurs because of changes in the con-text of reinforcement, which in turn producesvarying degrees of response inhibition. Thus,for a given component in which the reinforce-ment schedule is held constant, the excitatoryeffects of reinforcement will be constant, butthe degree of inhibition will vary as a functionof the weighted average of the reinforcementin the situation. Moreover, the context of re-inforcement will vary depending upon whichcomponent is present at any given time, andthe effects of reinforcement variation on theweighted average will depend on the locus ofvariation. At issue is whether such an ap-proach can provide an accurate description ofthe interactions that are actually observed. Thepresent study addressed this issue by investi-gating the effects of parametric variations ofreinforcement rate in a three-component mul-tiple schedule.

METHODSubjects

Four White Carneaux pigeons were main-tained at 80% of their free-feeding weights byadditional feeding, when necessary, followingexperimental sessions. All subjects had exten-sive experience with multiple schedules in thesame apparatus as used here (see Williams,1981, Experiment 2).

ApparatusA standard conditioning chamber was con-

structed from a plastic picnic chest. Thechamber was approximately 30.5 cm in alldimensions. On the front panel were mountedtwo pigeons keys, 1.7 cm in diameter, whichrequired a minimum force of 0.10 N for op-eration. Only the left key was used, and theright key remained dark throughout the study.The stimuli were projected onto the rear ofthe key by a standard 28-V, 12-stimulus IEEin-line projector. Ten cm below the keys wasa 5- by 5-cm aperture through which the birds

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Table 1

Schedules of reinforcement and number of sessions for thevarious experimental conditions.

Schedule value (min) No. ofCondi- ses-tion Comp. A Comp. B Comp. C sions

1 VI 3 VI 6 VI 1.25 252 VI 1.25 VI 6 VI 3 203 VI 6 VI 1.25 VI 3 304 VI 3 VI 1.25 VI 6 305 VI 1.25 VI 3 VI 6 306 VI 6 VI 3 VI 1.25 307 VI 3 VI 3 VI 3 258 VI 6 VI 3 VI 6 259 VI 1.25 VI 1.25 VI 6 2510 VI 1.25 VI 1.25 VI 1.25 2011 VI 6 VI 6 VI 6 25

could eat when the food hopper was activated.No houselight was used, so the chamber was

completely dark except for the illuminationprovided by the keylight and the occasionalpresentations of the hopper light.

ProcedureA three-component multiple schedule was

used throughout training, with the fixed se-quence, ABC ABC, etc. The stimulus that ac-companied Component A was three horizon-tal white lines on a dark background; forComponent B, a diffuse green keylight; andfor Component C, a white circle on a yellowbackground. The schedules in effect duringeach component for each condition are shownin Table 1, along with the number of sessionsof training on each. The number of sessions,determined prior to each condition, was basedon experience with respect to how much train-ing would be necessary for responding of allfour subjects to stabilize. Additional training(in blocks of five sessions) was used on thoseoccasions when the responding in all threecomponents did not appear by visual inspec-tions to be stable. The various schedules were

constructed from the distribution described byFleshler and Hoffman (1962) with 18 differ-ent intervals for the VI 1.25- and 3-minschedules and 12 different intervals for the VI6-min schedule. Component duration was 10 s

throughout the experiment because previouswork has shown this to be the duration thatproduces maximal schedule interaction (Shimp& Wheatley, 1971). The component timerstopped during reinforcement (2.5-s access to

Purina pigeon chow). Each session was in ef-fect for 150 cycles, lasting about 75 min.

Data AnalysisThe parameters of the various equations

were fit using an iterative nonlinear regres-sion program that is part of the University ofCalifornia statistical package that is commer-cially available (Ralston, 1981). The stabilitycriterion for the fit was that the error variancenot decrease by more than 0.001% over fivesuccessive iterations.

RESULTSEvidence for Asymmetric InteractionsA major premise of the weighted-average

model (Equation 7) is that separate terms arerequired for the component schedule that pre-cedes and the component schedule that followsthe component of immediate concern, becausethose two sources of alternative reinforcementhave systematically differing effects on behav-ior. A previous study with the same subjects(Williams, 1981, Experiment 2) addressed thisissue directly, with the result that contrast ef-fects from the schedule that followed weresubstantially larger than those from the sched-ule that preceded (although the responding ofone subject, R-38, was affected substantiallyby both alternative sources of reinforcement).However, that study was conducted consid-erably prior to the present work, so it is im-portant to establish that similar differencesoccurred with the present procedure beforeproceeding to the application of Equation 7 tothe entire array of results.

Given the set of schedules shown in Table1, the best method of determining the differ-ential effects of the preceding versus followingschedules is to examine those conditions inwhich the schedule during one component washeld constant while those of the surroundingcomponents were interchanged between suc-cessive conditions. Response rates in the con-stant component can then be compared withrespect to whether the higher reinforcementrate occurred in the preceding or in the fol-lowing component. Five pairs of conditions of-fer this comparison: Conditions 1 versus 2, 3versus 4, and 5 versus 6, all with ComponentB as the constant schedule, and Conditions 2versus 3 and 4 versus 5 with Component Cas the constant schedule. For example, with


Conditions 1 versus 2 as the comparison, ahigher response rate in Component B duringCondition 2 indicates a larger effect of thefollowing schedule because the frequency ofreinforcement in the component schedule thatfollowed was lower in Condition 2 than inCondition 1. If reinforcement frequency in thepreceding component schedule had the greatereffects on behavior in Component B, responserate should be higher in Condition 1.

Figure 1 summarizes the results of the fivedyads relevant to the issue of contrast asym-metry by showing the proportion of the totalresponding in each pair of conditions duringthe condition in which the constant compo-nent was followed by the lower reinforcementrate (e.g., for Conditions 1 and 2, the propor-tion of total responding in Component B thatoccurred during Condition 2). Values greaterthan .5 indicate a greater effect of the schedulethat follows; values less than .5 indicate agreater effect of the schedule that precedes.Across the five comparisons, values greaterthan .5 were obtained in all five cases for Sub-jects R-75 and R-56, in four of five cases forSubject R-63, and in 3 of 5 cases for SubjectR-38. It should be noted that two of the threevalues below .5 occurred in the first compar-ison (Component B for Conditions 1 vs. 2), inwhich the effect of the preceding schedulemight be expected to be larger, because astrong local contrast effect at the beginning ofa component is more likely early in training(see Williams, 1983a, for a review). In gen-eral, therefore, the present data provide strongevidence for asymmetrical schedule interac-tions. It also should be noted that the differ-ences from .5 that are shown do not reflect theabsolute size of the effect of either the preced-ing or following schedule, but only the differ-ences between those effects. It is thus possiblefor both effects to be large, but for the differ-ence between them to be small and inconsis-tent. It is partly for that reason that a quan-titative description is desirable that representseach source of alternative reinforcement by aseparate term.

Evaluations of Equations that Predict AbsoluteRates of RespondingThe data shown in Table 2 were fit by

Equation 7 with a nonlinear regression pro-cedure. Four separate fits were made for datafrom each subject: one for response rates in

0.7

rn7S R-u r-u1 R-63SUBJECT

Fig. 1. Relative response rates for the unchangedcomponents for pairs of conditions in which the Iocationsof the surrounding schedules were interchanged. Theserelative rates were obtained by dividing the number ofresponses in the condition of each pair in which the fol-lowing component schedule had the lower reinforcementrate, by the sum of responses during both conditions ofthe pair. Each bar corresponds to a separate comparison,the order being Condition 1 versus 2 (Component B), 3versus 4 (B), 5 versus 6 (B), 2 versus 3 (C), and 4 versus5 (C).

each component, and one for the pooling ofresponse rates from all components. Thus, forfits to individual components there were 11data points (10 for R-75 because it became illbefore completion of the last condition),whereas for the fit to the aggregates there were33 (30 for R-75).

Table 3 shows the parameter values for eachseparate fit, along with the percentage of vari-ance accounted for. In general it is clear thatthe fits for the individual components werebetter than those for the aggregates. The me-dian percentage of variance accounted for bythe fits to the individual components was 91.5(mean = 88.7), whereas the correspondingpercentage for the aggregate fits was 83.1(mean = 82.9). The apparent reason for thisdifference was the variance in the s parame-ter, because there is prior evidence that suchvariation represents more than experimentalerror; that is, other research in our laboratoryhas shown consistent differences in the re-sponse rates maintained by different types ofstimuli even when correlated with identicalreinforcement schedules. Although such dif-ferences can be captured by different s valuesfor the fits to individual components, they areaveraged out for the fits to the aggregates, andthus contribute to error variance.The quality of the fits also was variable

across subjects. Excellent fits were obtained

51


Table 2Results for individual subjects on all conditions. Response rates are in terms of responses/minute. Reinforcement rates (obtained) are in terms of reinforcements/hour. Data are fromthe last five sessions of each condition. The numbers in parentheses below each response ratecorrespond to the standard deviation of the rates across the five sessions.

Con- Responses in Reinforcements in

di- component componenttion A B C A B C

Subject R-7516.4 67.1 22.1 9.0 47.7(1.5) (2.4)21.4 37.3 46.4 10.4 23.4(2.8) (1.1)46.2 50.3 9.9 41.4 23.0(6.4) (9.4)66.8 22.8 21.6 47.3 10.4(6.3) (4.2)43.0 16.9 46.8 20.7 10.4(8.1) (4.1)25.1 52.4 9.5 23.4 47.7(1.8) (4.5)34.5 37.0 22.1 22.5 27.9(4.9) (5.3)39.1 25.5 9.9 23.4 10.8(6.1) (5.1)66.1 15.1 38.7 49.5 11.3(10.2) (1.9)43.8 39.1 38.7 51.3 56.7(5.2) (4.1)

Subject R-5618.2 68.6 22.1 9.9 51.3(1.5) (1.4)21.0 49.1 45.9 9.0 23.0(2.9) (3.8)70.7 68.8 10.4 41.4 22.5(5.1) (5.2)74.2 38.4 22.5 46.8 9.9(1.9) (4.9)67.8 30.6 45.5 21.6 9.5(5.6) (4.1)45.7 67.4 9.9 22.1 50.0(3.1) (3.1)60.8 64.6 21.2 23.0 24.8(4.6) (7.4)66.8 52.1 10.8 22.5 10.4(3.5) (4.7)70.1 29.8 40.5 45.9 11.7(3.7) (3.8)50.7 63.7 41.9 45.9 55.4

(12.0) (2.6)45.1 53.0 12.6 11.3 11.3(1.6) (4.5)

Con- Responses in Reinforcements in

di- component componenttion A B C A B C

Subject R-381 36.6 31.6 88.4 21.2 10.4 51.8

(3.7) (1.9) (6.5)2 67.6 19.4 48.4 45.0 10.4 22.1

(3.2) (2.6) (1.4)3 29.9 87.2 63.4 10.1 44.2 23.6

(3.1) (7.8) (1.3)4 41.4 105.8 33.6 22.5 47.6 12.4

(2.5) (4.6) (1.1)5 61.6 46.2 15.9 47.3 23.0 9.5

(4.4) (2.1) (3.9)6 23.3 53.3 60.7 11.3 20.7 47.3

(3.0) (6.2) (4.5)7 38.0 54.6 44.0 22.5 21.7 25.7

(1.2) (5.9) (1.1)8 26.8 53.0 25.6 10.4 21.6 10.4

(2.9) (3.7) (2.7)9 55.5 82.6 19.5 40.1 49.5 10.8

(4.7) (6.5) (2.4)10 42.6 63.2 44.1 38.7 49.5 58.5

(4.3) (6.3) (7.5)11 22.7 32.2 27.3 10.8 10.8 11.3

(3.6) (2.6) (2.6)

1 54.9(2.9)

2 54.7(4.8)

3 25.8(2.1)

4 31.2(4.8)

5 46.7(1.8)

6 23.9(2.4)

7 34.1(4.6)

8 36.0(2.2)

9 38.8(3.6)

10 41.0(3.7)

11 32.0(2.8)

Subject R-6327.9 59.8 21.6 9.0 47.3(1.8) (4.6)19.6 43.2 46.8 10.8 23.4(3.8) (5.7)53.4 51.2 10.9 41.3 22.5(2.4) (4.2)64.2 27.6 22.5 39.6 9.5(5.4) (3.7)44.1 15.0 46.8 22.1 12.2(5.4) (3.7)40.3 46.0 9.9 23.0 47.7(6.0) (5.3)44.5 45.7 22.1 23.9 27.0(2.4) (2.6)59.0 43.1 11.3 24.3 11.7(2.3) (2.6)56.9 21.4 41.9 49.1 10.8(4.2) (5.3)50.2 43.4 41.0 49.5 59.9(5.0) (4.1)35.6 33.8 10.4 10.4 10.4(3.1) (3.6)

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1 45.5(3.4)

2 70.4(4.9)

3 20.7(3.9)

4 28.8(3.6)

5 54.2(5.1)

6 21.6(2.3)

7 28.7(3.2)

8 20.5(4.2)

9 35.7(5.6)

10 31.2(2.9)

11

1 49.8(5.4)

2 72.9(3.2)

3 33.8(5.2)

4 51.6(4.6)

5 72.4(1.7)

6 40.2(2.0)

7 62.4(7.9)

8 47.2(6.7)

9 61.1(1.8)

10 54.8(3.3)

11 47.9(2.6)


Table 3

Best fitting values for the parameters of Equation 7. Also presented is the percentage of varianceaccounted for by the best fit. Shown in parentheses below each parameter estimate is itsstandard deviation. Four fits are presented for each subject: one for each separate component,and one for all components pooled together.

Component Component

A B C All A B C All

R-75 R-56

s 42.4 66.9 41.4 49.3 65.2 59.7 59.8 61.3(11.0) (10.1) (11.3) (6.6) (5.4) (7.0) (3.7) (3.2)

p 0.15 -0.27 0.42 0.05 0.12 0.11 0.03 0.09(0.22) (0.14) (0.30) (0.13) (0.06) (0.11) (0.05) (0.04)

f 1.21 1.04 0.51 0.92 0.18 0.51 0.32 0.32(0.35) (0.21) (0.31) (0.19) (0.07) (0.15) (0.07) (0.05)

C 3.3 20.7 1.1 7.8 3.3 2.4 1.0 1.8(7.8) (7.2) (7.2) (4.6) (2.2) (2.8) (1.4) (1.3)

% Var. 93.1 97.9 84.6 88.5 89.6 92.2 94.8 88.4

R-38 R-63

s 74.7 97.4 63.3 76.5 45.5 53.8 45.1 47.7(13.8) (21.4) (22.8) (14.4) (5.4) (5.7) (6.9) (3.7)

p 0.29 0.93 0.28 0.42 -0.07 0.24 0.04 0.06(0.15) (0.59) (0.41) (0.24) (0.08) (0.12) (0.14) (0.07)

f 0.17 0.46 0.46 0.45 0.44 0.28 0.35 0.37(0.15) (0.41) (0.51) (0.24) (0.14) (0.11) (0.18) (0.09)

C 18.1 19.7 8.9 14.1 3.1 4.2 1.7 2.7(8.3) (10.1) (11.8) (7.5) (3.1) (2.9) (3.5) (1.9)

% Var. 93.0 92.9 74.0 77.0 83.7 90.9 77.7 77.8

for Subjects R-75 and R-56, with the per-centage of variance accounted for generallyover 90%. For Subjects R-38 and R-63, therewere good fits to some individual components,but in general the fits were considerably poor-er.Of major interest are the values of the var-

ious parameters. The s and C parameters aresimilar to those reported in the literature whenEquation 6 was fit to response rates main-tained by single VI schedules (dc Villiers &Herrnstein, 1976), although it should be not-ed that the value of s does not correspond tothe maximum response rate possible (the as-ymptote instead is 41 + p + f]). Of greatestinterest are the p and f values. For the 16different fits shown in Table 3, 14 have alarger value forf than for p, and in most casesthe difference is substantial. In terms of theirabsolute values, 14 of the 16 values of p arepositive, with a mean of 0.18, and all 16 val-ues of f were positive with a mean of 0.50.The fits are thus consistent with previousfindings (and Figure 1) showing that schedule

interactions are due predominantly to varia-tions in reinforcement rate in the componentschedule that follows. However, the consis-tently positive values of p also indicate somecontribution from the component schedule thatprecedes. The magnitude of this contributionvaried substantially across subjects, as onlyR-38 showed a strong effect. Previous datafrom the same 4 subjects showed a similarpattern (see Experiment 2 of Williams, 1981),suggesting that such individual differencespersist over substantial periods of time.

Fits of Relative Response RatesA major potential source of variability with

respect to absolute rates of responding wascondition-to-condition variation in asymptoticresponse rates. Such variation should not besurprising, given the long duration of the ex-periment (over 1 year), inasmuch as "organ-ismic" variables might change substantiallyover that length of time. This source of vari-ability can be reduced by using relative ratesof responding, on the assumption that organ-

53


ismic variables affect responding in each com-ponent equally. To do this, Equation 8 wasderived by dividing Equation 7 with respectto Bn by the same equation with respect toBn+1 (note that for Rn+I, the component sched-ule that followed was component n - 1, be-cause the three-component schedule recycledcontinuously):

SnRn + pRn-I + fRn+

BnBn+l

Sn+lRn+l

1 +p +fRn+l

+ pRn + fRn-1 +p +f

Rearranging terms:Bn Sn RnBn+1 Sn+1 Rn+1

Rn+1 + pRn + fRn-I + C + pC +fCRn + pRni1 + fRn+l + C +pC+fC'

and setting n = sI and C(1 + p + f) = C':Sn+1

Bn. Rn Rn+1 + pRn + fRn1 + C'Bn+1 Rn+1 Rn + pRn-1 + fRn+l + C'

(8)

The data shown in Table 2 were fit byEquation 8, once again using an iterative non-linear regression program. An initial difficultyin determining the best fitting parameter val-ues was that the values of p, f, and C tendedto be inversely correlated, so that large valuesof C (not C') occasionally occurred in con-junction with small values of p or f (or viceversa). To avoid this problem, the value of C(not C') was set constant at the value derivedfrom the fit of the aggregate of the absoluterates shown in Table 3 (e.g., for Subject R-75,C was set constant at 7.8). This added a de-gree of freedom to the curve-fitting procedureand also produced p and f values that wereinterpretable, with the results shown in thefirst column for each subject in Table 4.The pattern of parameter values obtained

with the fit of Equation 8 was similar to thatfor Equation 7 (see Table 3) except that theweights for p andftended to be slightly larger.Of the 16 separate fits of Equation 8, 15 es-timates ofp were positive, and all 16 estimatesof f were positive. The mean value of p was

0.24, and that of f was 0.58. The major dif-ference between fits of Equations 7 and 8 wasthat Equation 8 accounted for substantiallymore of the variance, despite having one fewerfree parameter. Considering only the fits tothe individual pairs of components (e.g.,A/B), the median percentage of variance ac-counted for was 96.5 (mean = 95.5) and inonly one case was it less than 90%. The fitsof the aggregates of all pairs of componentswere substantially worse, but this was ex-pected because of the considerable variabilityin s' with respect to the individual pairs ofcomponents; that is, s' for the fits to the ag-gregates was essentially the mean of the s'values for the individual component pairs, sothat any variation around that mean increasedthe error variance.

Although the estimates of both p andf wereconsistently positive, it remains to be deter-mined whether the inclusion of both alterna-tive sources of reinforcement contributed sig-nificantly to the functionally effective contextof reinforcement. In other words, would thefits of the data be significantly worse if one orthe other source of alternative reinforcementwere omitted from the equation? To make thisassessment, two additional equations were fitto the data. Equation 9 is analogous to Equa-tion 7 except that the term for the componentschedule that precedes has been omitted;Equation 10 is similar except for omission ofthe term for the component schedule that fol-lows. Thus,

BnS}M.+CRnB= Rn +JRn+l+

1 +Jand

(9)

(10)Bn=S Rn

Rn +pRni1 + C1 +p

The corresponding expressions for relative re-sponse rates are then derived by writingEquations 9 and 10 for both components nand n + 1, with Equations 11 and 12 as theresult (where C' = C[ 1 + f] for Equation 11and C[1 + p] for Equation 12):

Bn ,IRn Rn+I+fRn-1 +CBn+1 Rn+1 Rn +fRn+l + C'

and

54

Rn

I


Bn , Rn Rn+1 + pRn + C'Bn+1 Rn+1 Rn + pRn-I + c

The fits provided by Equations 11 and 12are shown in the second and third columnsunder each subject's listings in Table 4. Ingeneral, the quality of the fits was substan-tially worse than that provided by Equation8, which included both alternative sources ofreinforcement. When the term for the com-ponent schedule that precedes was removed(Equation 11), the mean percentage of vari-ance accounted for dropped from 95.5 to 88.0,although this difference occurred primarilybecause of only 2 subjects, R-38 and R-63.,With removal of the term for the componentschedule that follows (Equation 12), the dropin variance accounted for was considerablygreater, from 95.5 to 67.9%. Once again Sub-ject R-38 was notably different from the other3 subjects, as the change in quality of the fitto its data was approximately equal wheneither the preceding or following componentschedule was removed. For the remaining 3subjects, the term for the following componentschedule was more important, as the reductionin the percentage of variance accounted forwas substantially greater when its corre-sponding term was removed.

Because Equation 8 has one more free pa-rameter than Equations 11 or 12, and the ad-dition of any free parameter will always in-crease the variance accounted for, it isimportant to establish that the better fits pro-vided by Equation 8 represent more thansimply the absorption of error variance. Thedifference in variance accounted for by Equa-tion 8, versus either Equation 11 or Equation12, was thus tested statistically by an F testas given by Equation 13 (Cohen & Cohen,1975, pp. 135-136) in which R,2 refers to theproportion of variance accounted for by Equa-tion 8, R22 to the proportion of variance ac-counted for by the alternative equation (eitherEquation 11 or 12), k1 to the number of in-dependent parameters associated with Equa-tion 8, k2 to the number of independent pa-rameters associated with the alternativeequation, andN to the number of observationsbeing fitted. The degrees of freedom corre-sponding to the numerator and denominatorof the F ratio is then given by kI - k2 andN - k, - 1, respectively. Thus,

F- (R12 - R22)/(k1 - k2)(1 - R12)/(N - k-1)

(13)

The resulting F values are shown in Table5, with those significant at the .05 level ofconfidence indicated by asterisks. All but oneof the comparisons with the following com-ponent schedule excluded were reliable,whereas 10 of the 16 comparisons with thepreceding component schedule excluded weresignificant. All of the differences involving fitsto the aggregate data (that had greater powerbecause of the larger number of degrees offreedom) were reliable, which implies thatboth the preceding and the following schedulecomponents provided significant increments tothe quality of the description of the relativeresponse rates, and thus contributed indepen-dently to the context of reinforcement.

DISCUSSIONRelativistic Effects of ReinforcementThe central issue addressed here is whether

interactions in multiple schedules can be char-acterized adequately by a relativistic concep-tion of reinforcement that regards the re-sponse-strengthening effect of any absoluterate of reinforcement to be inversely relatedto the "context" in which it occurs. A criticalassumption in testing this conception was thatthe context can be conceptualized as theweighted average of the various sources of re-inforcement available in the situation-in thepresent case, the prevailing component of themultiple schedule, the component that justpreceded, and the component that consistentlyfollows. As noted in the introduction, the useof different weights seems to be demanded byprevious research showing that the componentschedule that consistently follows is a morepotent source of behavioral contrast than isthe component schedule that precedes (Wil-liams, 1981). The present analysis is consis-tent with those previous findings: For 3 of the4 subjects, the empirically determined weightsfor the following component schedule wereconsiderably larger than those for the preced-ing component schedule. For the remainingsubject, the contributions of the two alterna-tive sources of reinforcement were compara-ble.The validity of the weighted-average con-

55

56 BEN A. WILLIAMS and JOHN T. WIXTED

Table 4Best fitting values for the parameters of Equations 8, 11, and 12. Also presented is the per-centage of variance accounted for by the best fit. Shown in parentheses below each parameterestimate is its standard deviation. Four fits are presented for each subject, with the relevantresponse ratios indicated under the component column.

Eq. 8 Eq. 11 Eq. 12 Eq. 8 Eq. 11 Eq. 12

R-75 R-56

0.99 1.15(0.12) (0.23)

0.31(0.25)

0.69(0.24)91.8 65.10.91 1.25(0.13) (0.32)

0.36(0.33)

1.34(0.48)94.8 65.81.04 1.20

(0.05) (0.13)0.13(0.09)

0.40(0.07)97.4 77.10.99 1.21(0.07) (0.13)

0.26(0.12)

0.78(0.17)90.6

R-38

65.6

1.07(0.06)0.21(0.05)0.63(0.10)97.40.91(0.05)0.09(0.03)0.44(0.06)97.6

1.02(0.05)0.03(0.03)0.28(0.06)93.2

1.00(0.05)0.09(0.04)0.43(0.07)87.7

1.11 1.19(0.13) (0.22)

0.34(0.21)

0.54(0.15)85.8 54.01.02 1.09(0.05) (0.11)

0.26(0.16)

0.45(0.07)95.0 59.01.02 1.09

(0.05) (0.11)0.12(0.07)

0.30(0.06)92.1 56.21.02 1.11(0.06) (0.10)

0.23(0.08)

0.44(0.07)84.2

R-63

50.4

0.91 0.81(0.19) (0.10)

0.96(0.42)

0.44(0.40)65.9

1.31(0.1 1)

0.44(0.13)94.1

1.18(0.12)

90.71.42(0.17)0.28(0.18)

85.81.14

(0.09)0.27(0.16)

0.86(0.06)0.38(0.09)0.59(0.13)96.0

1.12(0.11)-0.05(0.06)0.48(0.12)89.7

1.04(0.04)0.01

(0.03)

0.94 0.94(0.13) (0.15)

0.43(0.22)

0.41(0.16)73.2

1.12(0.11)

0.45(0.12)88.9

1.04(0.04)

65.5

1.25(0.22)0.20(0.16)

48.81.11(0.11)0.12(0.07)

Comp.A/B

Comp.B/C

Comp.C/A

ALL

St

p

f

% Var.St

p

f

% Var.St

p

f

% Var.St

p

f

% Var.

0.91(0.07)0.27(0.09)0.94(0.25)97.50.91(0.1 1)0.17(0.13)1.32

(0.47)96.2

1.04(0.05)0.03(0.03)0.39(0.08)97.60.96(0.07)0.14(0.06)0.82(0.17)92.7

Comp.A/B

s'

p

f

% Var.Comp.B/C

p

f

0.71(0.07)1.45

(0.56)1.08

(0.57)96.9

1.24(0.09)0.18(0.09)0.45(0.13)96.7

1.09(0.10)0.26(0.1 1)

Comp.C/A

% Var.st

p


Table 4 (Continued)

Eq. 8 Eq. 11 Eq. 12 Eq. 8 Eq. 11 Eq. 12

R-38 R-63

f 0.14 0.19 0.31 0.32(0.14) (0.15) (0.05) (0.05)

% Var. 91.8 82.3 90.4 95.1 95.0 56.9ALL st 1.00 1.11 1.14 1.02 1.03 1.11

(0.07) (0.09) (0.09) (0.06) (0.06) (0.10)p 0.43 0.43 0.09 0.23

(0.13) (0.13) (0.04) (0.08)f 0.53 0.46 0.41 0.42

(0.19) (0.16) (0.08) (0.07)% Var. 89.5 79.3 82.6 84.5 82.1 52.8

ception of context of reinforcement can be as-sessed only indirectly by the present data, byexamining how well the data are described byequations derived from that concept. In gen-eral, the quality of the description was high,especially for relative response rates, with over95% of the variance accounted for. Thus, interms of the criterion of goodness of fit, therelativistic conception of reinforcement wasstrongly supported. Moreover, all of the pa-rameters of the equations are theoreticallymeaningful, and the ranges of all values thatwere derived for those parameters were con-sistent with the corresponding conceptual in-terpretations.An important implication of the present

weighted-average conception of context ofreinforcement is that it argues strongly againstviewing the matching of relative response ratesto relative reinforcement rates as the limitingform of multiple-schedule interactions. Thusit suggests a basic difference between multipleand concurrent schedules. Some previous re-

search provides evidence that matching does

occur in multiple schedules with short com-ponent durations (Merigan, Miller, & Gol-lub, 1975; Shimp & Wheatley, 1971; Todo-rov, 1972; Williams, 1983b), but otherresearch challenges the generality of thosefindings (Charman & Davison, 1982). All suchprevious research has used two-componentmultiple schedules, which are incapable of re-vealing any asymmetry in the effects of com-ponents that precede and components that fol-low (inasmuch as they are always the samecomponent). The fact that the two effects arefunctionally different causes serious difficultyfor any conception of matching as a generallaw of schedule interactions, because suchasymmetry implies that matching cannot oc-cur generally with multiple schedules withthree or more components. That is, matchingimplies that relative response rate producedby a particular relative reinforcement rateshould be invariant, whereas the asymmetri-cal interactions imply that the effect of anygiven relative reinforcement rate will dependcritically upon its location in the sequence of

Table 5

F values for the statistical test given by Equation 13. F values which exceeded the .05 level ofconfidence are indicated by asterisks.

Eq. 8 vs. Eq. 11 Eq. 8 vs. Eq. 12

Component ComponentSubject A/B B/C C/A All A/B B/C C/A All

R-75 13.68* 2.21 0.50 7.48* 77.76* 48.0* 51.25* 96.52*R-56 31.23* 7.58* 1.13 8.25* 116.86* 112.58* 38.09* 87.94*R-38 70.0* 5.52 8.11* 28.17* 14.00* 23.12* 1.20 19.06*R-63 39.90* 0.54 0.14 4.49* 53.58* 27.8* 54.57* 59.31 *

57


components. The same limitation also appliesto any use of relative rate of reinforcement asa molar variable that does not include a func-tional separation of the different sources ofreinforcement context (e.g., the generalizedmatching law: Equation 1).

Extensions to Two-Component SchedulesAn important question regarding the gen-

erality of the weighted-average context modelis how well it can be applied to previous stud-ies of multiple schedule that have used onlytwo components. With such schedules, thecomponents that precede and those that followare always identical, so the two effects can nolonger be considered separately. This combi-nation can be captured by allowing p + f =m, so that the corresponding equations for ab-solute and relative response rates in two-com-ponent schedules become Equations 14 and15, respectively:

Bn SRn + mRn+l ( 14)R1+ mR+ + c (41 +m

andBn St Rn .Rn+1 + mRn + C' (15)Bn+I Rn+1 Rn + mRn+, + C'

Equation 15 was fit to the data of the threeprevious studies of two-component multipleschedule that involved short component du-rations and that varied relative rate of rein-forcement over several values. The results areshown in Table 6 along with fits by Equation1 for comparison. Both equations were fit withthe same nonlinear regression procedure asthat used with the present results. As occurredin the analysis of the relative response ratesdescribed in the Results section, the fit ofEquation 12 was complicated by the correla-tion between the value of m and C, such thatthese occasionally assumed absolute values thatwere so high as to be uninterpretable. Con-sequently, the value of C (not C') was con-strained between the values of 0 to 20 rein-forcers per hour.

For the experiment by Shimp and Wheat-ley (1971), the data analyzed were all of theconditions involving component durations ofeither 2, 5, or 10 s. For the study by Charmanand Davison (1982), the data analyzed weretaken only from their Experiment 2, which

held component duration constant at 5 s.Shown separately for that study are their con-ditions in which the two components of theschedule were presented on the same or dif-ferent keys. Separate conditions involving oneversus two response keys were also studied byMerigan et al. (1975), but only the two-keyconditions are presented here because thoseauthors reported very weak interactions whenonly a single key was involved. It should benoted that Merigan et al. varied relative du-ration of reinforcement, but the other twostudies varied relative rate of reinforcement.Of primary interest from Table 6 is the

relative quality of the descriptions provided bythe two equations. They appear comparablefor the study by Shimp and Wheatley (1971),as both account for more than 90% of the vari-ance for all subjects. They are also compara-ble for both conditions of the study by Char-man and Davison (1982), although bothaccounted for substantially less of the vari-ance. It also should be noted that the degreeof schedule interaction (captured by m and a)was also substantially less in that study. Thetwo equations do differ, however, with respectto the fits of the data of Merigan et al., inwhich the generalized matching law (Equa-tion 1) provided a more accurate descriptionfor all subjects. In general, therefore, Equa-tion 15 provides an excellent description of theresults from previous two-component multipleschedules (at least those with brief componentdurations), although the generalized matchinglaw provides a more accurate description un-der some circumstances. As noted in the in-troduction, however, the generalized matchinglaw provides no account of changes in abso-lute response rates and has no method of de-picting the asymmetric effects of the compo-nent schedules that precede and that followthe one under consideration when more thantwo-component schedules are employed.

Problems with the Context ModelDespite the accuracy of description provid-

ed by the present model, several features ofthe analysis presented in the Results sectionoffer reservations about its conceptual ade-quacy. As shown in Tables 3 and 4, there wasconsiderable variability in the obtained valuesof the three parameters, particularly for p andf Such variability implies that the size of thecontrast effects depended upon characteristics


Table 6Best fitting parameter values for Equations 15 and 1 when fit to previous studies using two-component multiple schedules. The number of data points fit for each subject is shown inparentheses next to the study citations.

Equation 15 Equation 1

Subject S' m C % var. b a % var.

Shimp & Wheatley 13 0.91 0.78 0 91.0 0.87 .80 94.5(9) 15 1.00 1.04 0 92.9 1.07 .94 97.8

20 1.05 1.08 0 96.3 1.16 .91 95.1Charman & Davison 151 0.57 1.32 0 80.4 1.07 .61 81.4(one-key) (5) 152 0.97 0.15 20 83.1 0.95 .48 91.7

153 0.94 0.49 0 94.6 0.95 .49 92.8154 1.10 0.19 20 54.1 1.07 .44 69.3155 0.96 0.25 0 90.6 0.98 .29 81.9156 0.90 -0.11 20 88.7 0.91 .27 81.2

Charman & Davison 151 1.57 0.40 20 98.1 1.35 .90 87.6(two-key) (5) 152 0.44 0.47 0 89.2 0.43 .59 93.7

153 1.25 0.31 0 64.0 1.15 .48 74.3154 0.91 0.14 20 92.0 0.87 .36 86.8155 1.59 0.25 0 83.0 1.55 .37 89.2156 1.75 0.11 20 95.9 1.78 .39 92.3

Merigan et al. 1 0.94 0.26 0 85.1 0.92 .35 93.2(6) 2 0.98 0.96 0 77.6 0.95 .88 94.5

3 1.11 0.76 0 93.0 1.09 .81 97.94 0.99 0.80 0 91.8 0.96 .84 96.2

of the particular stimuli that accompanied aschedule, which seems somewhat implausible.One possible explanation for such effects isthat the pairs of components differed with re-spect to the degree of stimulus similarity,which has been shown to affect the degree ofcontrast in previous studies (Blough, 1983;White, Pipe, & McLean, 1984).A more troublesome feature of the param-

ter estimates was the large variation in thevalues of C across the various components, andthe large degree of error associated with thoseestimates (see the standard deviations of C inTable 3). Moreover, fits of the relative re-sponse rates when C was allowed to vary free-ly often produced uninterpretable results. Asimilar problem occurred in the analysis ofprevious two-component studies (see Table 6);because it was necessary to constrain the val-ues allowed for C, and the estimates of C thatwere produced ended up at one or the otherextreme of the constraint boundaries. In fact,for the majority of subjects C assumed a valueof zero, which implies that it contributednothing to the model's fitting of the data. Andbecause positive values of C represent an es-sential feature of the model's derivation (seeCatania, 1973), the implication is that the

conceptual rationale of the model may be ill-founded. However, the problem with the es-timates of C did not occur when Equation 7was used to describe absolute response rates,as the estimated values of C were all positiveand similar to those obtained from fits to datafrom single VI schedules (de Villiers &Herrnstein, 1976). This suggests that the dif-ficulties associated with the estimates based onfits of relative response rates may have moreto do with the method of analysis than withthe conceptual basis of the model per se. Apossible reason is that in the original deri-vation of the model (see the introduction), theparameters s and C are correlated, and a sim-ilar but inverse correlation should also occurbetween C and m, given any error in mea-surement, because they are combined into theparameter C' when Equation 15 is fit to rel-ative response rates. That is, the curve-fittingprogram produced an estimate of C', and thevalue of C was determined algebraically bysubstituting in the values of m. Any error inthe estimate of m would then produce an op-posite corresponding error in the value of C,which could be nonsensical in terms of themodel.The role of measurement error also has im-

59


plications for the criticism of the context mod-el offered by Charman and Davison (1983).They demonstrated that increasing food de-privation decreased the degree of interactionin a multiple schedule, as indexed by the valueof a in Equation 1. They then argued thatEquation 5 could not account for this changebecause neither the value of C nor m changedsystematically as a function of body weight.Moreover, the estimated values of m from theirdata typically were substantially greater than1.0 and the values of C were typically nega-tive. Their method of obtaining these esti-mates was by an algebraic solution for m andC using Equation 15 for two separate rein-forcement ratios for each body weight. Unfor-tunately, such an algebraic solution assumesno error in measurement. When such errorsdo occur, as they inevitably do, large system-atic distortions in the estimates of the param-eters may occur.To demonstrate the role of measurement

error, we used the obtained reinforcement ra-tios from the data of Charman and Davison(1983) to generate idealized response ratiosfrom Equation 15. For purposes of this sim-ulation, m was assumed to be 0.5 and C was7.0 (reinforcers/hour). By definition, there-fore, the context model fit the data perfectly.These idealized response ratios were then sys-tematically distorted by introducing a 10% biastoward one of the two components of the mul-tiple schedule (i.e., all relative response ratioswere multiplied by 1.10). These distorted re-sponse ratios were then substituted back intoEquation 15 and the values of m and C weredetermined algebraically by the same methodused by Charman and Davison. The resultwas a pattern quite close to that reported bythose authors: m values were typically greaterthan 1.0 and C values were negative. Giventhat the original error-free data were gener-ated by the model, the implication is that theparameter estimates provided by the algebraicsolutions cannot be taken as serious evidenceagainst the model. Whether a similar argu-ment can be used to account for the inconsis-tencies in the parameter estimates noted above(particularly with respect to C in Table 6)remains to be determined.

Comparisons with Alternative ModelsWhatever problems remain to be resolved

with respect to the weighted-context model, a

major argument in its favor is that it is theonly quantitative description yet proposed thatappears capable of handling the major qual-itative features of behavioral interactions ob-served in multiple schedules. As noted in theintroduction, the generalized matching lawprovides no basis for accounting for changesin absolute response rates, including the mostsalient feature of those changes-behavioralcontrast. Similarly, Herrnstein's (1970) ex-tension of his treatment of concurrent sched-ules to multiple schedules is also deficient forreasons cited in the introduction. The onlyremaining account that has been advanced isthat based on the notion of behavioral com-petition, first proposed independently byHenton and Iversen (1978) and by Hinsonand Staddon (1978) and developed quantita-tively by McLean and White (1983) and byStaddon (1982). Accordingly, response rate inone component of a multiple schedule is notdirectly affected by the reinforcement rate inthe alternative component. Instead, changes inresponse rate are due to variation in the de-gree of competing behavior within that partic-ular component. Changes in response rate thatostensibly occur because of changes in alter-native reinforcement are then assumed to bedue to the re-allocation of the competing be-havior patterns between components.

Although the behavioral-competition ap-proach offers an attractive alternative to thepresent account based on "context of rein-forcement" as a primitive variable, it facesseveral major difficulties. Not only does it pro-vide no account of the asymmetrical interac-tions seen in the present study, but its accountof contrast in any conventional multipleschedule is based on hypothetical changes inunobserved competing behavior that suppos-edly mediates the observed changes in re-sponse rate. It thus seems unlikely that anyquantitative description of the observed be-havior will be possible based only on the in-dependent variables manipulated by the ex-perimenter. But perhaps most importantly,behavioral-competition models cannot accountfor the major finding concerning multiple-schedule interactions-that contrast does notdepend upon the behavior maintained in thealternative component but, instead, dependsonly on the alternative rate of reinforcement(see Williams, 1983, for a more extensivetreatment of this problem). In contrast, the

60

ANALYSIS OF CONTRAST 61

present account provides a quantitative de-scription of both absolute and relative re-sponse rates, based entirely on observed in-dependent variables and tied to a majorconceptual rationale about response strength(cf. Catania, 1973; Keller, 1980). The accu-racy of that description and its ability to in-corporate the asymmetrical effects of differ-ence sources of reinforcement context arguestrongly in its favor.

REFERENCESBaum, W. M. (1974). On two types of deviation from

the matching law: Bias and undermatching. Journal ofthe Experimental Analysis of Behavior, 22, 231-242.

Blough, P. M. (1983). Local contrast in multiple sched-ules: The effect of stimulus discriminability. Journal ofthe Experimental Analysis of Behavior, 39, 427-435.

Catania, A. C. (1973). Self-inhibiting effects of rein-forcement. Journal of the Experimental Analysis of Be-havior, 19, 517-526.

Charman, L., & Davison, M. (1982). On the effects ofcomponent durations and component reinforcementrates in multiple schedules. Journal of the ExperimentalAnalysis of Behavior, 37, 417-439.

Charman, L., & Davison, M. (1983). On the effects offood deprivation and component reinforcer rates onmultiple-schedule performance. Journal of the Experi-mental Analysis of Behavior, 40, 239-251.

Cohen, J., & Cohen, P. (1975). Applied multiple regres-sion/correlation analysis for the behavioral sciences.Hillsdale, NJ: Erlbaum.

de Villiers, P. A. (1977). Choice in concurrent schedulesand a quantitative formulation of the law of effect. InW. K. Honig & J. E. R. Staddon (Eds.), Handbook ofoperant behavior (pp. 233-287). Englewood Cliffs, NJ:Prentice-Hall.

de Villiers, P. A., & Herrnstein, R. J. (1976). Towarda law of response strength. Psychological Bulletin, 83,1131-1153.

Fantino, E. (1977). Conditioned reinforcement: Choiceand information. In W. K. Honig & J. E. R. Staddon(Eds.), Handbook of operant behavior (pp. 313-339).Englewood Cliffs, NJ: Prentice-Hall.

Farley, J. (1980). Automaintenance, contrast, and con-tingencies: Effects of local vs. overall and prior vs.impending reinforcement context. Learning and Moti-vation, 11, 19-48.

Fleshler, M., & Hoffman, H. S. (1962). A progressionfor generating variable-interval schedules. Journal ofthe Experimental Analysis of Behavior, 5, 529-530.

Gibbon, J. (1981). The contingency problem in auto-shaping. In C. M. Locurto, H. S. Terrace, & J. Gib-bon (Eds.), Autoshaping and conditioning theory (pp.285-308). New York: Academic Press.

Henton, W. W., & Iversen, I. H. (1978). Classical con-ditioning and operant conditioning: A response patternanalysis. New York: Springer-Verlag.

Herrnstein, R. J. (1970). On the law of effect. Journalof the Experimental Analysis of Behavior, 13, 243-266.

Herrnstein, R. J. (1974). Formal properties of the

matching law. Journal of the Experimental Analysis ofBehavior, 21, 159-164.

Hinson, J. M., & Staddon, J. E. R. (1978). Behavioralcompetition: A mechanism for schedule interactions.Science, 202, 432-434.

Keller, K. J. (1980). Inhibitory effects of reinforcementand a model of fixed-interval performances. AnimalLearning & Behavior, 8, 102-109.

Lander, D. G., & Irwin, R. J. (1968). Multiple sched-ules: Effects of the distribution of reinforcements be-tween components on the distribution of responses be-tween components. Journal ofthe Experimental Analysisof Behavior, 11, 517-524.

McLean, A. P., & White, K. G. (1983). Temporal con-straint on choice: Sensitivity and bias in multipleschedules. Journal of the Experimental Analysis of Be-havior, 39, 405-426.

McSweeney, F. K. (1980). Differences between rates ofresponding emitted during simple and multiple sched-ules. Animal Learning & Behavior, 8, 392-400.

Merigan, W. H., Miller, J. S., & Gollub, L. R. (1975).Short-component multiple schedules: Effects of rela-tive reinforcement duration. Journal of the Experimen-tal Analysis of Behavior, 24, 183-189.

Ralston, M. (1981). PAR derivative-free nonlinearregression. In W. J. Dixon (Ed.), BMDP statisticalsoftware (pp. 305-314). Los Angeles: University ofCalifornia Press.

Shimp, C. P., & Wheatley, K. L. (1971). Matching torelative reinforcement frequency in multiple scheduleswith a short component duration. Journal of the Ex-perimental Analysis of Behavior, 15, 205-210.

Spealman, R. D., & Gollub, L. R. (1974). Behavioralinteractions in multiple variable-interval schedules.Journal of the Experimental Analysis of Behavior, 22,471-481.

Staddon, J. E. R. (1982). Behavioral competition, con-trast, and matching. In M. L. Commons, R. J. Herrn-stein, & H. Rachlin (Eds.), Quantitative analyses ofbehavior: Vol. 2. Matching and maximizing accounts (pp.243-261). Cambridge, MA: Ballinger.

Todorov, J. C. (1972). Component duration and rela-tive response rates in multiple schedules. Journal of theExperimental Analysis of Behavior, 17, 45-79.

White, K. G., Pipe, M-E., & McLean, A. P. (1984).Stimulus and reinforcer relativity in multiple sched-ules: Local and dimensional effects on sensitivity toreinforcement. Journal of the Experimental Analysis ofBehavior, 41, 69-81.

Williams, B. A. (1976a). Behavioral contrast as a func-tion of the temporal location of reinforcement. Journalof the Experimental Analysis of Behavior, 26, 57-64.

Williams, B. A. (1976b). Elicited responding to signalsfor reinforcement: The effects of overall versus localchanges in reinforcement probability. Journal of theExperimental Analysis of Behavior, 26, 213-220.

Williams, B. A. (1979). Contrast, component duration,and the following schedule of reinforcement. Journalof Experimental Psychology: Animal Behavior Processes,5, 379-396.

Williams, B. A. (1980). Contrast, signaled reinforce-ment, and the relative law of effect. American Journalof Psychology, 93, 617-629.

Williams, B. A. (1981). The following schedule of re-inforcement as a fundamental determinant of steady

62 BEN A. WILLIAMS andJOHN T. WIXTED

state contrast in multiple schedules. Joumnal of the Ex-perimental Analysis of Behavior, 35, 293-310.

Williams, B. A. (1983a). Another look at contrast inmultiple schedules. Journal ofthe Experimental Analysisof Behavior, 39, 345-384.

Williams, B. A. (1983b). Overmatching in multipleschedules. Behaviour Analysis Letters, 3, 171-182.

Received September 1, 1984Final acceptance October 13, 1985

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